Chebyshev's Theorem

The Russian mathematician P. L. Chebyshev (1821- 1894) discovered that the fraction of observations falling between two distinct values, whose differences from the mean have the same absolute value, is related to the variance of the population. Chebyshev's Theorem gives a conservative estimate to the above percentage.

For any population or sample, at least (1 - (1 / k)^{2}) of the observations in the data set fall within k standard deviations of the mean, where k ³ 1.

Using the concept of z scores, we can restate Chebyshev's Theorem to say that for any population or sample, the proportion of all observations, whose z score has an absolute value less than or equal to k, is no less than

(1 - (1 / k^{2} )). For k = 1, this theorem states that the fraction of all observations having a z score between -1 and 1 is (1 - (1 / 1))^{2} = 0; of course, this is not a very helpful statement. But for k ³ 1, Chebyshev's Theorem provides a lower bound to the proportion of measurements that are within a certain number of standard deviations from the mean. This lower bound estimate can be very helpful when the distribution of a particular population is unknown or mathematically intractable.

EX Chebyshev's Theorem can be utilized for the following values of k:

k = 1.5 1 - (1 / 1.5^{2}) = 0.5556 of all observations fall within 1.5s of m.

k = 2.0 1 - (1 / 2.0^{2}) = 0.7500 of all observations fall within 2.0s of m.

k = 2.5 1 - (1 / 2.5^{2}) = 0.8400 of all observations fall within 2.5s of m.

k = 3.0 1 - (1 / 3.0^{2}) = 0.8889 of all observations fall within 3.0s of m.

## Comments

November 16, 2010 - 23:02.